Gameboard Ramsey Numbers

Sequences Bn(p,q) of connected parts of Euclidean and hyperbolic (p,q)-mosaic graphs are considered. The smallest n such that any 2-coloring of the edges of Bn( p,q) contains a given monochromatic graph G is introduced as gameboard Ramsey number rp,q(G). For p ≥ 4 it is proved that these Ramsey numbers exist for finitely many graphs only. For p = 3 there exist infinitely many numbers r3,q(G). For p ≥ 6 all gameboard Ramsey numbers are determined.